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In a preceding paper 'On rings of operators, III" J. von Neumann has developed a theory of normed operators for operators in an arbitrary factor 9R.2 In part the procedure is as follows: First [[A]], the norm of A, is defined for all A e tR of finite rank (Definition 1.3.1) and fundamental sequences of such operators and an equivalence relation for fundamental sequences are defined (Definitions 1.4.1, 1.4.2 and 1.4.3). Then the key theorem (Theorem III) is proved that every fundamental sequence is strongly convergent operatorially, that two fundamental sequences converge strongly to the same limit if and only if they are equivalent, and finally that the norms of a fundamental sequence converge numerically to a limit depending only on the operator limit of the sequence. The proof of Theorem III makes use of two important lemmas: LEMMA 1.4.1. If An, n = 1, 2, * * is a fundamental sequence with [[AJ]] + 0 as n -oo then An converges strongly to 0 as n -* ??. LEMMA 1.4.3. If An, n = 1, 2, * * is a fundamental sequence with An converging strongly to 0 as n -* , then [[An]] -0 as n + oo. The purpose of the present note is to point out how the proofs of these two lemmas can be simplified by using more fully the results of the paper 'On rings of operators, II' by Murray and von Neumann.3 We require the following lemma. LEMMA. If OR is a factor in a finite case and An E OR with An converging strongly to O as n -oo , then [[An]] -*0 as n + oo . PROOF. There is a finite set of elements gi, * * , Mg,, such that [[An]]2 = Tr9 (A*An) = ET-, (A*Angi, gi) = 2t7l (Angi, Angi) = o l| Ang 112 for all n.4 Since A gi -* 0 as n oo for all i, and m is finite, the lemma follows. PROOF OF LEMMA 1.4.3. Suppose the lemma false, if possible. Then for some e > 0 we could assume that [[An]] > e for all n, by choice of a suitable subsequence. For some fixed no , [[An Ar]] < e/2 for all n, r _ no. From the definition of a fundamental sequence, Ano is contained in some closed linear manifold 91 of finite dimensionality relative to 'DR. Let E be the projection on W1.

The object of this note is to fill a gap in the definition of, and to make certain remarks concerning a group, hr, which was introduced in H. G. (p. 421).1 The definition should read as follows: Let r be a given group, whose elements we denote by small greek letters, and let H be an aggregate of individuals a, a-', b, b-1, *, such that a # a-', b 5 bV' etc.2 Let h(H) C F be a single-valued, but not necessarily (1 1), transformation of H in F, such that h(a-') = {h(a) }-'. We denote by hr the group generated by all pairs (a, i), to be denoted by at, with (a, 7 ' = (a-', ), subject to the relations (1.1) atbt,= brat, for each a, b C H, i, il C F, where r = th(a)t7 'n (if a = b, t = l, this reduces to at = ath(a)). We recall from H. G. the definitions of the automorphism Oi(hr) = hr. defined by

Introduction. The purpose of the present paper is to derive some properties of a ring (in short: I-ring) in which the chain conditions on the right ideals (see ?1) hold concerning its automorphisms. A special case of this problem arises by the study of a normal product of two algebras A and B over a field F.' If A is the normal and B the supplementary factor, then A possesses a basis a,, a2, ... , an over F such that aiB = Bai, i = 1, * * *, n. These relations define automorphisms of the algebra B, and it is of importance to study the behaviour of B under these automorphisms in order to know the structure of the normal product. First we consider in this note a finite or infinite set M of automorphisms of a I ring R. If K denotes the kernel of R (i.e. the set of all the elements which remain invariant under each automorphism which belongs to M), then we prove that the quotient-ring K/AT, where N is the radical of K, is semi-simple-i.e. the chain condition holds. We then assume that M is a finite group and that na = 0 for a * 0 , where a e R and n is the order of the group M. In this case we prove that the kernel of a I ring R is a potent ring if, and only if the ring R is potent. There exists a one to one correspondence between the primitive right ideals of K and the normal-primitive right ideals of R, i.e. the minimal potent right-ideals of R which remain invariant under each automorphism of R belonging to M; further, the radical of the kernel is a subset of the radical of R. We further prove that the kernel of a semi simple ring is also a semi simple ring. Finally, a new proof is derived for the theorem which states that the group algebra of a finite group over a field F is semi simple. For a detailed study of the notions used in this paper, consult B. L. v. d. Waerden, Moderne Algebra, II, pp. 149-172.

In this paper we will study a class of groups of even order which satisfy the following condition: (TI): two different Sylow 2-groups contain only the identity element in common. There are three series of finite non-abelian simple groups known to satisfy the above condition (TI). Let q denote a power of 2 greater than 2. The linear fractional group in 2 variables over the field of q elements, denoted here by L2(q), satisfies the condition (TI). The projective unitary groups U3(q) provide another series of groups satisfying (TI). The third series consists of groups G(q) defined by the author in [7]. The main result of this paper is the converse of the above statement. We will prove the following theorem.

In a recent paper (Math. Ann. vol. 114, p. 237), Jonas has given a very interesting new property of W-congruences. After some general remarks, this paper will give here new, easy, and purely projective demonstrations of these theorems. On the last page we state a new problem which seems to be important for the theory of W-congruences. We shall say that two surfaces, whose points are in a given one-to-one reciprocal correspondence, are congruence-transforms of each other if they are the focal surfaces of the congruence generated by the straight lines joining two corresponding points of the two surfaces. We shall denote by x, y, *. the points whose homogeneous projective coordinates are xi, yi, .** (i = 1, 2, 3, 4), by ax + by + * ** the point whose coordinates are axi + byi + * i i . If (p is a function of two parameters u, v, we denote by so", so,,, #uu, Or".,, ... the derivatives 8co/cu, 8cp/8v, l2V/8u2, 8 2V/8U8V, ... .When xi are functions of u, v, we denote by x", X, * the points whose coordinates are 8xi/8u, 8xi/8v, * . By (x, y, z, t) we denote the determinant

In this paper we shall study some problems which are purely settheoretical in character, but which arose in connection with a problem in the theory of models. The principal results of this paper form an important part of the proofs of the series of results in the theory of models which are announced in [1], [2], [3], and in the appendix of [4]. The present article, however, is entirely free of any notions from the theory of models. It is well-known that, for every infinite cardinal a, there exist nonprincipal prime ideals in the field S(X) of all subsets of a set X of power a, and furthermore there are exactly 221 such prime ideals. The object of this paper is to show that non-principal prime ideals with certain additional properties exist. Let X be a set of power a, and let S( Y) be the set of all non-empty finite subsets of a set Y. If f, g map S.( Y) into the field of subsets of X, we write g > f if, for all a E S.( Y), g(a) -f(a). Let us say that an ideal I in the field S(X) is SJ( Y)-good if, for every monotonic function f on S,(Y) into I, there exists an additive function g > f on S,( Y) into I. If Y is countable, then every ideal I is S.( Y)-good (cf. Corollary 4.2). Our two main results are as follows.

(1.4) Q(n) ns r2s(n) __ (-))! (e) nO( s -> 3, M It=1 ~~(s-i!m where e is any positive integer and se (s) is 1, 0 or 2s according as Q is odd, e 2 mod 4 or e = 0 mod 4. For --1 this says that the function nl-sr28(n) is in the mean (on the average) equal to ITS/(s 1)! if s > 3. For other values of e the formula implies a weighted asymptotic average of the same function, the weight factors being re(n) for varying n. These are therefore suitable functions to smooth out on the average the irregularities of the function n1ls r2S(n). What is proved is that the irregularity is smoothed out in the limit as mn becomes infinite; some empirical evidence (not recorded in the paper) indicates that the smoothing effect probably is generally in evidence for rather small values of m. The paper also contains results similar to (1.4) concerning representations of integers as sums of an even number of triangular numbers and also

Introduction. In a complex Kn a simplex ois said to be regular if its star has the homology characters of an n-cell, the star being defined as the set of all simplexes of K having o-, as a face. If all the simplexes are regular K is said to be a manifold. This paper results from investigations of irregularity and of questions concerning the extent to which the irregularity of a simplex implies the irregularity of its faces. Answers are given to two main problems around which interest has crystallized: (i) Can K have all its simplexes regular except some of dimension p? (ii) Can K have all its simplexes regular except some of dimension > p? The second question arises because the answer to the first is in the negative unless p = 0 or n 1, and it can itself be answered in the affirmative; this can be seen by constructing an actual example for p = n 1, and by further subdividing it to introduce irregularity in lower dimensions. Lastly we make use of the technique of the earlier part to solve a problem left open in Tucker's thesis.' He finds three conditions on a cell-complex that it should be a manifold, closure, star, and intercept uniformity; but it is not there proved that intercept uniformity is independent of the other two. We construct an example of a complex having star and closure uniformity but lacking intercept uniformity, and thereby establish the required independence. The terminology of the paper is largely that of Lefschetz ;2 however, the term 'complement' is used instead of 'link complex'. I am indebted to Professor Tucker for the suggestion of the problems treated here, and also for some of the ideas involved in their solution.

In an earlier paper [1, ?4]2 the term invariant complex wN-as applied to any subset II of the elements of a finite quasigroup G such that (Ha) (Hb) for all a, b takes the form Hc for some element c in G. It was shown that a necessary and sufficient condition for the existence of a given homomorphism of G is that G contain an invariant complex H. MAore specifically, a homomorphism partitions the elements of G into disjoint subsets (complexes) H1 , H2 , such that any one of them may be denoted by H and then [1, Theorem 4.14] a-' of them, H included, take the form Hp for suitably chosen elements p of G. These complexes, under the rule of operation (Ha) (Hb) = He constitute a quasigroup (the quotient quasigroup G/H) isomorphic with the homomorph of G. Conversely, an invariant complex has the property that for its cosets Hx we have Ha = Hb or Ha A Hb = 0 so that, if p < Ha, the mapping p -* Ha constitutes a homomorphism of G. In view of the preceeding remarks, it is clear that a study of the relations H < K (proper inclusion of H by K), H A K = D, or HK = M between invariant complexes H and K is fundamental to the theory of homomorphisms of G. In [1, ??4.4, 4.5] such a study was undertaken, but with restrictions on H and K of the type (Hx) (Hy) = H(xy) or, still stronger, H(Ha) = Ha for all a. In Section 2 of the present paper the above relations are studied without the restrictions mentioned. If D t 0, D and M are shown (Theorems 2.1 and 2.4) to be invariant under no stronger hypothesis than that H and K are invariant. Theorem 2.2 shows the implications of H < K and Theorem 2.3 those of Hp A Kq # 0 for cosets of H and K. In Section 3 properties of maximal invariant complexes are studied in the light of Section 2. One of the principal difficulties encountered is the fact that none, one, or more than one of the cosets of an invariant complex may be a subquasigroup. This means that the analogy between invariant complexes of quasigroups and invariant subgroups of groups is not as close in Section 3 as in Section 2. However, if it be required, as it is in Section 3.2, that the quasigroup contain at least one idempotent element b considerable similarity of results remains. For example, Theorem 3.3 on composition series which terminate with the idempotent element b is an analogue of the Jordan-Holder Theorem. For a loop (a quasigroup with a two sided identity element e) introduced by A. A. Albert [2] there is only one idempotent element, namely e. The theory

1. One aspect of the theory of addition of independent random variables is the frequency with which the partial sums change sign. Investigations of this nature were originated by Paul L6vy, in a paper [1] which contains a wealth of ideas. This problem as such was mentioned by Feller in his 1945 address [2]. In the case where the partial sums can actually vanish the problem falls under the head of 'recurrent events', a general theory of which was recently developed in a paper by Feller [3]. A very special case had been studied in detail by Hunt and myself [41. Generalizing the problem in a natural way we shall consider the number of -times Tn with which the sequence of reduced partial sums Sk E(Sk), k = 1, 2, * *, n crosses a given value c. We shall establish the limiting distribution of T. in the case where the random variables have a common distribution with a finite third absolute moment.2 In all cases the limiting distribution is that of the 'positive normal', but the proper normalizing factor depends on the case. A distribution function is said to be of the 'lattice type' if and only if it is a step function with all its discontinuities located at the multiples of a certain number. Without loss of generality we may suppose that this number is an integer. The minimum distance between two discontinuities is called the 'span'. A complication arises when the given value c is a possible value of Sk E(Sk). In the case ,of a non-lattice distribution this makes no difference but in the case of a lattice distribution it does. In fact, the meaning of the phrase 'Sk E(Sk) crosses the value c' becomes ambiguous and the definition used in Theorem 2 must be regarded as merely convenient. However, other possible definitions are subject to the same treatment. We leave open the problem of obtaining strong limit theorems for T. ; presumably they must be of the same form as those in [4]. We denote the kth moment of F(x) by ak, the kth absolute moment by fik. For brevity we write a for a, ,and ,72 for a2 a, .

The background of this problem is developed in an earlier paper.' We here complete the reduction referred to on page 518 as the "third problem." That is, we here determine conditions on the coefficients of a positive quaternary quadratic form which define a reduced form. This reduced form has the property that in every class of equivalent forms there is one and only one reduced form. In other words, every form is equivalent to one and only one reduced form. These conditions are in the form of linear inequalities on the coefficients and fall into two groups. First, there are inequalities which are completely symmetric, that is, are independent of the numbering of the variables of the form. These conditions are, except for those mentioned in the previous paper, of a distinctly new type and possess many obvious advantages one of which is the brevity with which the conditions may be stated. Second, there are the usual inequalities which control permutations of the variables. In the reduction, no distinction is made between proper and improper equivalence. That is, transformations of determinant -1 are used as freely as those of determinant +1. However if one desires a set of reduced forms of given determinant having the property that every form of that determinant is properly equivalent to one and only one form of the set, the set may be found as follows. First, find the set of reduced forms of the given determinant by the inequalities of this paper. Then for each reduced form which has no automorph of determinant -1 we will have another form obtained from it by, for instance, changing the signs of the coefficients of xIx2, x1x3 and X1X4 in the form. Hence the number of forms in what we might call the properly reduced set will be the number of forms of the reduced set plus the number of forms in that set without automorphs of determinant -1. In the table of automorphs at the close of the paper, the automorphs of determinant -1 are starred. S. B. Townes, has, in an unpublished Chicago dissertation (1936), carried through the Eisenstein reduction for positive quaternary quadratic forms. Though it has the advantage of giving a leading coefficient which is the minimum of the form, etc., the proof is very much longer and the conditions on the form are more complex. The comparative shortness of this proof is due in large part to the symmetric conditions above referred to. Furthermore, the minimum of any reduced form of this paper can be obtained quickly.

1. This is a continuation of my paper [7] with the same title, which will be referred to as D', and which has to be supplemented in the manner described below in Appendix I. Indeed, the present paper is nothing else than a combination of the ideas of D' with the method of variation of structure, as applied to special types of semisimple groups by Calabi and by Calabi-Vesentini in their recent work'. We shall mainly be concerned with semisimple groups without compact components, i.e., with connected semisimple Lie groups having no connected compact normal subgroup. However, we begin by considering any connected Lie group G having at least one discrete subgroup with compact quotient; this implies that G is unimodular (it would even be enough for this to assume that G has a discrete subgroup H such that G/H has a finite measure, for the measure determined on G/Hby a right-invariant measure on G). Let n be the dimension of G; choose a basis X1, * , Xn for the space of right-invariant vector-fields on G; we have

Because of the continuous group operations, a topological group has some peculiar homotopy properties. It is well-known that the fundamental group of a topological group is commutative, [4, p. 54].2 The higher homotopy groups of a topological group have been investigated by W. Ifurewicz [11], B. Eckmann [5], and others. According to J. H. C. Whitehead [13], an arcwise connected topological group is n-simple for each n > 1 in the sense of S. Eilenberg [6]. According to R. H. Fox [81], the torus homotopy groups of a topological group G are all commutative; then it follows that the Whitehead products, [14], between the elements of the homotopy groups of G are always zero. The object of the present paper is to generalize these results to the class of special homogeneous spaces, defined in ?3, including the spheres Sn as particular cases. With an aim to do this, we shall give in ?2 some additional homotopy properties of a topological group. Our main results3 are embodied in the theorems of ?3. Throughout this paper, let G be a locally connected topological group. Since the components of G are homeomorphic to each other, we assume G to be arcwise connected. Let II be an arewise connected closed subgroup of G. A mapping is a continuous transformation, and a homotopy is a continuous family of mappings ft, (0 < t ? 1).

In this paper we study holomorphic functions from the point of view of function algebras. By a function algebra we mean an algebra A of continuous (complex-valued) functions on a compact Hausdorff space X which is closed in the sup norm, 11 * Ix. We assume also that A contains the constants and separates the points of X. In this case, it is well-known, the space of maximal ideals S(A) of A is a compact Hausdorff space in the weak topology determined by A; A can be represented as a closed algebra of continuous functions on S(A), and X can be embedded as a closed subset of S(A) [11]. Further, for any f e A, I I f I x = I I f I 1s(,). Any subset B of S(A) such that for all f eA, Iif lB = lIfIIS(A) is called a boundary for A. The intersection of all closed boundaries is a boundary, and is called the Silov boundary, denoted by 1(A) [12]. The Silov boundary is not necessarily the smallest boundary for A; the latter need not even exist. But if A is separable (as a Banach space), then S(A) is metric, and this minimal boundary does exist. In this case 1(A) is just the closure of the minimal boundary [4]. Let M be a complex analytic manifold, and let K be a compact subset of M. Let H(K) be the algebra of all functions holomorphic in a neighborhood of K, and let A(K) represent the closure of H(K) in the norm II IlK. How can we determine S(A(K)), 1(A(K)), and, as it makes sense in this case, since A(K) is separable, the minimal boundary? These questions have been discussed for certain types of compact sets in Cn (complex n-dimensional vector space) by K. de Leeuw [10], and K. Hoffman [14]. The papers of S. Bergman on distinguished boundary domains [3] are the first to indicate the significance of the Silov boundary in several complex variables (see also D. Lowdenslager [17] and H. Bremermann [8]). The Silov boundary is the smallest subset of K on which we can hope to represent holomorphic functions by an integral formula. If M is of dimension one (i.e., a Riemann surface), then K is the space of maximal ideals, and OK is the Silov boundary of A(K) [1, 4, 22]. In higher dimensions, neither of these is in any sense generally true. For example, if 470

In 1921, S. Mazurkiewicz [1] raised the question: if a bounded continuum in a space of m dimensions is topologically equivalent to each of its nondegenerate subcontinua, is it an arc? In 1948, making use of a certain plane indecomposable continuum, E. E. Moise [2] showed that this question may be answered in the negative. In this paper it is shown that for compact decomposable continua in a metric space, the answer is affirmative. At the outset the space S is assumed to be compact and metric. After Theorem 5 it will be convenient to place further restrictions on S. An arc is the homeomorphic image of the number interval [0, 1]. A continuum is said to be decomposable if it is the sum of two continua neither of which is M. An arc is, of course, decomposable. If a continuum is not decomposable, it is said to be indecomposable. A decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua will be called an A-continuum. (The terminology in this paper is similar to that used in R. L. Moore's Foundations of Point Set Theory [3]). If x and y are distinct points of a continuum M, some subcontinuum of M is irreducible from x to y [3, Theorem 34, Ch. I]. Hence if M is an A-continuum, M itself must be irreducible between some two of its points.

Let f: Ni Pi (i = 1, 2) be Co mappings of Co manifolds. We will say fi and f2 are isomorphic if there exist Co diffeomorphisms h: N1 N2 and h': P1 P2 such that h' o f1 = fo h. The main notion studied in this series of papers will be that of (oo -structurally) stable mapping. Roughly speaking, a Co mapping f: N P is said to be (ao -structurally) stable if every Co mapping g: N o P which is sufficiently close to f in a suitable topology (the direct limit of the fine Ck topologies, as k Ao) is isomorphic to f (cf. Thom and Levine [7]). There is an infinitesimal notion corresponding to stability. Namely, we say a Co mapping f: N o P is infinitesimally stable if for every Co vector field w along f there exist Co vector fields u on N and v on P such that w = Tf a u + v o f. (Our terminology and notation is the following: TN denotes the tangent bundle of N. N, denotes the tangent space to N at n, for each n e N. A vector field along f is a mapping w: N TP such that w(n) e Pf(,) for all n e N. The mapping of tangent bundles induced by f is denoted Tf: TN-+ TP.) In Stability of Mappings: II, we will show that a proper, infinitesimally stable, Co mapping is stable. In a later paper we will show that a proper, stable, Co mapping is infinitesimally stable. Thus, a proper, Co mapping is stable if and only if it is infinitesimally stable. A very simple example will be given in Stability of Mappings: II to show that infinitesimal stability of f does not imply stability of f if the hypothesis that f is proper is dropped. Also, if this hypothesis is dropped, stability of f does not imply infinitesimal stability. B. Morin has constructed an example which shows this. The purpose of this paper is to state and prove an analytic result which will be used in our proof that a proper, infinitesimally stable, Co mapping is stable. The result is that there exist mappings Q and H satisfying a certain identity (equation (1) of ? 2) and having certain "differentiability" properties.

S. Smale, using his theory of handlebodies, has classified, under diffeomorphism closed, simply connected, smooth 5-manifolds with vanishing second Stiefel-Whitney class. C.T.C. Wall has given a classification of (n 1)-connected (2n + 1)-manifolds which does not however cover the case n = 2. In this paper we complete the classification of simply connected 5-manifolds. A.A.Markov has proved that a general classification of 5-manifolds is impossible, but it seems reasonable to hope for results in the case of 5-manifolds with a given fundamental group. The second Stiefel-Whitney class of a simply connected manifold may be regarded as a homomorphism w: H2(M; Z) Z, and we may arrange w to be non-zero on at most one element of a 'basis' (0.5), this element having order 2i for some i (Lemma C). Then i is a diffeomorphism invariant i(M) of M. If H2(X) -H(M), and i(X) = i(M), where X and M are simply connected 5-manifolds, then there are (0.8) isomorphisms 0: H2(X) H2(M) which preserve the linking form b on the torsion subgroups (0.7), and which satisfy w(M)oO = w(X). The basic theorem (2.2) states that any such isomorphism may be realized by a diffeomorphism of X with M. Thus H2(M) and i(M) form a complete set of invariants for the diffeomorphism classification. On the other hand b imposes restrictions on the second homology group (Lemma E), and hence on the decomposability of the manifolds. Using results of C.T.C.Wall on diffeomorphisms of 4-manifolds, it is possible to construct an example of an indecomposable manifold for each possible homology group (? 1) and, using these, to give a canonical manifold in each diffeomorphism class (Theorem 2.3). In addition to the main theorems, ?2 contains some corollaries and applications of them. The manner of construction of the indecomposable manifolds and manifolds similar to them produces minimal handle decompositions and allows the computation of embedding and immersion dimensions. The nature of the invariants also allows an extension of the results. The proof of Theorem 2.2 is omitted from ? 2 and occupies the remainder of the paper. X and M as above are necessarily cobordant (Lemma F), and the first step is to find a cobordism with minimal homotopy groups (? 3), i.e., one which is simply connected and with second homology group zero or, if iw(X) # 0, Z2. In ?4 modifications are described of which one enlarges the

This paper has two main purposes. In the first place, we shall generalize Brouwer's original definition of degree [2] for a mapping f: Mn Qn of one nmanifold into another to what we shall call the twisted degree. Geometrically the new definition is as elementary and intuitive as the old (see (3.8) below); it agrees with it for mappings of orientable manifolds and is more natural and gives a great deal more information in the non-orientable case. We shall denote the twisted degree of f simply by deg f. Part A is devoted to this definition. Secondly, we shall continue the program inaugurated by Brouwer [3] at the International Congress of Mathematicians in 1912 when he expressed his belief that in many cases the degree of a mapping would characterize its homotopy class, i.e., that two mappings of one orientable n-manifold M' into another Qf would be homotopic if and only if they had the same degree. He proved this for the mappings of an n-sphere S' into itself and subsequently [4] for the mappings of a 2-manifold into the 2-sphere. Hopf [11] then proved Brouwer's assertion for the case where M' is arbitrary and Qf = S' is the n-sphere; if M' is nonorientable the mod 2 degree is used here instead of the degree. Later the work of Hurewicz [13c] showed that the theorem remains true if S' is replaced by any n-manifold Qf for which the homotopy groups irr(Q') = 0 for r < n; (of course, S8 may be the only such n-manifold). We shall extend these results to the case where M' is arbitrary and ir,(Q') = 0 for 1 < r < n. It is immediately apparent that the degree alone is not enough here; something is required which will reflect algebraically the 1-dimensional portion of the homotopy. For this purpose, however, another simple invariant is at hand, namely the homomorphism (or homomorphism-class, see (3.2)) 0 of the fundamental group of Mn into that of Qf induced by f. Our principal theorem, Theorem II (see also Theorem I), asserts that for orientable M" and Qf these two invariants, the Brouwer degree and the induced 0, do characterize the homotopy class of f completely. Indeed, replacing the Brouwer degree by our twisted degree, the result holds also for mappings of non-orientable manifolds M' and Qf provided that the homomorphism 0 is "orientation-true" (see definition (3.1)). We show furthermore that if 0 is not orientation-true then (Theorem III) there are in any case not more than two homotopy classes for each 0; these are in some cases distinguished by the twisted degree, in some cases not, The detailed statements of all of these results, for both homotopy relative to a fixed point and free-homotopy, are given in Part B of the paper. Finally, we give a general theorem (Theorem IV) on the homotopy-types of n-manifolds whose homotopy groups between 1 and n vanish, All proofs are deferred to Part D.

Since the publication in 1956 of John Milnor's fundamental paper [l ] in which he constructs differentiable structures on S7 nondiffeomorphic to the standard one, several further results concerning differ­ entiable structures have been obtained by Milnor, R. Thorn, and others. This paper unifies and extends some of these results within the framework of an obstruction theory. Two differentiable manifolds ikf, N (connected, not necessarily compact) will be said to be combinatorially equivalent if they possess isomorphic C2 triangulations. If M and N are diffeomorphic, then they are combinatorially equivalent [5 ] ; we seek a partial converse. Let ƒ denote a linear isomorphism between C2 triangulations of the w-manifolds M and N; we attempt to redefine ƒ in neighborhoods of the open simplices of M, beginning with dimension n — 1 and working down, so as to make ƒ differentiable . After one step, ƒ is no longer a linear isomorphism between C2 triangulations ; hence one must formu­ late more general conditions for the induction hypothesis of this stepby-step procedure. The homeomorphism ƒ : M—>N is called a diffeomorphism mod L, where L is the w-skeleton of a C2 triangulation of M, if the following conditions are satisfied : (1) ƒ is a C2 diffeomorphism on each closed simplex of L. (2) ƒ* is one-to-one on the tangent vectors to L. (3) The subdivision of M is fine enough that for each simplex a of L, there are coordinate neighborhoods of a and/(cr) in which they are flat. (4) ƒ is of class C1 on M—L, with Df bounded and \Df\ bounded away from zero on any subset of M—L having compact closure. (Df is the Jacobian matrix.) (5) Let or be a simplex of L. Choose a coordinate neighborhood of â in which it is flat ; let z denote coordinates in the plane of

I will state the theorem which is the subject of this paper: If I is a bounded plane continuum possessing only one complementary domain and f is an orientation preserving homeomorphism of the plane leaving I invariant, then I will contain a point fixed under f. This theorem was first proved by Cartwright and Littlewood in [1] and since Hamilton and Newman have obtained an as yet unpublished proof based on the Brouwer plane translation theorem. It is the object of this paper to give a further proof, followed by some comments on the case of an orientation reversing transformation. The substance of this proof is contained in a paper [2] of L. E. J. Brouwer in 1910 in which that author gives a proof of the fixed point theorem for a "circular continuum" (which in fact appears to be a circle, at any rate it is locally connected). Let us suppose that f has no fixed point on I. Then there will exist b > 0 such that every point of I moves a distance at least 2b under f. We may then choose E > 0 such that every point within - of I moves a distance at least b under f. We now need a construction which involves two lemmas. LEMMA l. There exists a polygon H, whose interior complementary domain Hi contains I and lies within - of it while every point of H can be joined to some point of I by an arc in H of diameter less than b. Consider the domain D1 consisting of those points which are within 2E of I. Only a finite number of interior complementary domains of D1 can contain

1. The permutability of two bounded self-adjoint operators T1 and T2 defined on a Hilbert space S& is defined in the most natural and obvious manner: We say that T1 commutes with T2 if T17T2 = T2T1 throughout S. If {E1(X) } and {E2(0-) } are the canonical resolutions of the identity of T7 and T2 respectively, then the following result is well known: A necessary and sufficient condition that T7 commutes with T2 is the validity of any one of the following conditions: (a) The product T1T2 or T2T1 is self-adjoint. (b) E1(X) commutes with E2(0-) for each X and o. If Ti is a bounded self-adjoint operator and T2 any (not necessarily bounded) self-adjoint operator, then one usually defines permutability as follows (cf. [2] p. 31 or [4] p. 404): T1 commutes with T2 if Ti T2 (7 T2T1 . That is, if and only if, whenever f is in the domain of T2, Tif is also in the domain of T2 and T71T2f = T2T1f. This definition is equivalent to (b). Therefore, the permutability of any two selfadjoint operators is usually defined by (cf. [2], p. 50). DEFINITION 1. Two self-adjoint operators T1 and T2 with the corresponding canonical resolutions of the identity {E1(X)} and {E2(0-) } respectively are said to be permutable (or T1 commutes with T2) if E1(X) commutes with E2(o-) for each X and a-. The objective of this note is to show that two self-adjoint operators T1 and T2 commute if there exists a self-adjoint operator T such that T ( 7 T1T2 or T a T2T71. For the standard results and techniques of general Hilbert space theory used in this paper the reader is referred to the various papers of J. von Neumann [3], [4] and the books by M. H. Stone [5] and B. v. Sz. Nagy [2]. 2. To elaborate on our statements in ?1 we first recall the definition of multiplication of unbounded operators. The product AB of two operators A and B is defined as follows: (AB)f is defined if and only if Bf and A (Bf) are both defined and is then equal to the latter. DEFINITION 2. We say that the operators S, T are C' if S and T are self-adjoint, D(T) z D(S) (D((T) denotes the domain of T) and for f, g E D((T), the inner product (Sf, Tg) is Hermitean symmetric in f and g. LEMMA 1. Suppose that S, T are C' and E is a projection operator which commutes with T and such that R(E) ( D((T) (R(E) denotes the range of E). Let Si and T1 be the restrictions of ES and T respectively to R(E). Then S1 and T1 are bounded self-adjoint operators and S1 commutes with T1 . PROOF. Let f, g E R(E). Since R(E) C D((T) C D(S), f and g belong to D(S) and (I-E)Sf and (I-E)Sg belong to S& E R(E). Therefore

In [1], [2] and [3] no chain conditions occur anywhere; the strongest "finiteness condition" which had to be postulated for certain parts of the theory was that the rings have nilpotent radicals. (Square brackets refer to the references; [1], [2] and [3] will again be referred to as CPI, CPII and CPIII.) In the present paper we apply the results of CPI, II, III to completely primary rings with chain conditions. See the introduction of CPI for a short introduction to the present paper. All notations and terminologies are the same as for CPI, II, III. We remind the reader of the most important ones. Ring always means commutative ring with unit element 1 and zero element 0; otherwise, ring elements are denoted by lower case Greek letters, while a lower case German letter always denotes an ideal. The radical of a ring A, that is the ideal of its nilpotent elements, is denoted by N(A). The residue class ring A/N(A) is denoted by A and if N(A) has a finite exponent, this exponent is also called the exponent of A. If A C B are two rings where a C A and b C B, the extension of a in B is denoted by a* and the contraction of b in A by b* . (See CPI, sections 6a and 8 for the notions of extension and contraction of ideals.) If a = (a*)* for all a C A, we say that the (e, c)-condition is satisfied in the ring extension A C B. (See CPIII, Section 5, where the (e, c)-condition was formulated only for completely primary rings.) The quotient of two ideals a and a' is denoted by a: a'. An element of a ring which is not a divisor of zero, is called a regular element. The symbols C

The purpose of this paper is to give full descriptions of the examples announced in [8]. Let n be an integer, n > 2, and p a prime. Let AP denote the p-adic group (see below). Our principal result is the following. Example. There exists a compact metric space XI of dimension n and an action of AP on XI such that dim XI/Ap = n + 2. Such examples are interesting in relation to the important conjecture: If G is a compact group which acts effectively on a (generalized) manifold X, then G is a Lie group. This conjecture is proved for dim X ? 2 in [3] with generalizations given in [1], [9] and [10]. If this conjecture is false, then [4] there must be an example in which G = AP, and in this case [12], [2] dim X/Ap = n + 2. In this sense, the examples discussed here represent first approximations to counter-examples to the conjecture. The approximation is poor inasmuch as the spaces X" contain "large" n-dimensional sets; e.g., the n-skeleton of a triangulated (n+ 2)-manifold. Nor do we know of examples of AP action in which the dimension goes up by 3. It is known [12], [2] that the cohomology dimension cannot go up by more than 3. The authors expect to have more to say on these subjects. The reader might find the example of Kolmogoroff3 [5] a good introduction to the present paper. Kolmogoroff constructs a light open map which raises dimension by one; although not noted in [5], this map is the projection of an action of A2. One can construct analogous examples for all primes p, which as in [5], are based on the non-dimensional-full-valued spaces of Pontrjagin [7]. (A description of the Kolmogoroff example will appear in [11] as an illustration of the "A functor" of ? 2.) The additional complication in the examples described below stems from the following: If the dimension of a compact metric space X is to be decreased by 1 upon the removal of a subset D, D can be chosen to be totally disconnected. In contrast, to achieve a decrease of 2, D must have large components. Thus the individual modifications used in [7] are

If L is a subcomplex of a finite complex K, the topologically invariant factor group of the group of p-cycles of L bounding in K taken with respect to the subgroup of those bounding in L is useful in topology. The principal aim of this paper is to find its character group and to interpret the character group geometrically (theorems 4.2 and 4.4 at the end). Incidentally some algebraic and topological side results are obtained which will, it is hoped, help in finding character groups of less simple topologically invariant groups. The set of elements of a group containing more than one element are turned into a Hausdorff space by definition of open sets which satisfy, besides the Hausdorff axioms, axioms connecting the group operation and open sets, for instance those given by van Kampen' or Pontrjagin.2 A neighborhood of an element of the group is an open set containing the element, and a nucleus is a neighborhood of the identity. The closure of a set F of a Hausdorff space will be written F. A Hausdorff space is bicornpact if every covering of the whole space by open sets has a finite subset which covers the whole space (A.H. p. 863). The space will be called discrete if every element is an open set. As all groups here considered are Abelian, the group operation will be written additively, a + b. If H is the group of continuous homomorphisms of the topological Abelian group G into (A.H. "in") the additive group R of the real numbers mod 1 with the usual topology, then H is called the character group of G. When G is discrete and H is topologized as in v.K. or P.B., H is bicompact. When G is bicompact, H is given discrete topology. If G and H are topological Abelian groups and to every g E G and h E H there is an element gh = hg of R such that gh is a continuous function of g and of h, (g + g') h = gh + g'h and g(h + h') = gh + gh', then G and H are said to be paired; each h defines a continuous homomorphism of G into R, and each g a continuous homomorphism of H into R. If G and H are paired and S& is a subgroup of H, (M = (G, A) is defined as the set of all elements g E G such that gh = 0 for every h E Sg. The set (G, A) is a subgroup of G. (P.B. Note that none of the results in this paper depend on the Weyl-Peter theorem, see P.G.4)

In this paper we prove some theorems concerning 3-manifolds with boundaries. Any such set can be triangulated (Theorem 5). Furthermore, any locally tame closed subset of such a set is tame (Theorem 9). Working independently, Moise has subsequently obtained these results and is including his proofs along with other results in [5]. Much of the terminology used in this paper resembles that used in [4]. An n-manifold is a separable metric space K such that each point p of K lies in an open set that is topologically equivalent to En (Euclidean n-space). An n-manifold with boundary is a separable metric space K such that each point p of K lies in an open set N such that N is topologically equivalent to I' (a cube plus its interior in En). We note that an n-manifold is an n-manifold with boundary but not conversely. The term "boundary" is used in two senses. In the point set sense we say that the boundary of a set X is the intersection of the closure of X and the closure of the complement of X. However, if K is an n-manifold with boundary, we use Bd K to denote the points of K which do not have a neighborhood topologically equivalent to En. For example, if in a triangulated 3-manifold M with boundary, St(v) represents the star of a vertex v (the sum of all simplexes having a vertex at v) then Bd St(v) will be larger than the point set boundary of St(v) if v belongs to Bd M. In this paper we always use Bd(K) to denote the set of points of K which do not have a neighborhood topologically equivalent to En. The interior of an n-manifold K with boundary is K Bd K and is denoted by Int K. A set K can be triangulated if there is a homeomorphism h carrying a geometric complex C onto K. (We suppose that complexes are locally finite.) Let T(K) denote the collection of all subsets o of K such that ais the image under h of a simplex of C. If (a, xi, a2 x2, ... , ai xi) represents a point p of C

This, the problem of the least quadratic non residue, has often been investigated. The best result is due to Vinogradov, who proved that (1) n(k) = O(k1I(2Ve) where n(k) denotes the least positive quadratic non residue of the prime k. Also, several interesting results about n(k) have been obtained by A. Brauer through elementary methods. It has been conjectured that n(k) ko(E). This was first proved by Linnik on the assumption of the extended Riemann Hypothesis (E.R.H.). Recently, with the same hypothesis, S. Chowla and P. Erdds have improved Linnik's result to n(k) = O(exp ((log k)i+e)) In this paper I prove on the basis of the E.R.H. (2) n(k) = O((log k)2) S. Chow-la has proved that there exist infinitely many primes k where the first cl log k residues (mod k) are all quadratic residues. Hence, the upper bound on (2) cannot be improved beyond O(log k). The two results, the upper and lower bounds, are now not too far apart. The best possible bound for n(k) is probably not the bound in (2). One COuld(1 improve the bound in (2) by methods in this paper if one could derive any non trivial estimate in terms of k on

In his paper Reduced Powers of Cohomology Classes (7) N. E. Steenrod defines a set of invariant cohomology operations. These "reduced powers" include as a special case the squares which Steenrod introduced earlier (6). As the definition of the reduced powers depends essentially on a transformation group in the product space K X K X ... X K (p factors) of a complex K it seemed likely that at least in certain cases a close relation between his operations and the Smith-Richardson theory of symmetric products would exist. (See (3), (4) and (5).) In this paper we will discuss the case p = 2, and obtain a definition of the Steenrod squares directly from the exact sequences which are associated with the symmetric product of a space. Our main result can be stated as follows: given a complex X, set K = X X X and let k be the space obtained from K by identifying points which correspond to each other under the involution which exchanges the factors of K. Let 1 be the image in k of the diagonal of K. The Smith theory gives rise to a homomorphism A*:Hf(K; I2) -* H'(k, 1; I2). Further, the cohomology analogue of the Smith-Richardson construction of a basis in Hn(k, 1; I2) leads to the following result: there exists a canonical isomorphism 4' of the direct sum Gn = Z7n' Hi(X; I2) into H2 (k, 1; I2) with A*H2n(K; I2) in its image. Under this isomorphism the element {at}, at = Sqiu; (i = 1, ... , n) u E H'(X), of Gn maps onto A*(u X u). It follows that the Steenrod squares can be defined as the components of the element A*(u X u) in the direct sum decomposition of A*Hf(K) defined by 4,.1 I wish to take this opportunity to thank Professor Steenrod for his helpful remarks and encouragement. 1. Let K be a finite cell complex, t:K -* K an involution (5) of K (i.e. a homeomorphism with period 2 of K onto itself with the following properties: (a) t maps cells onto cells. (b) If a cell is mapped onto itself by t it remains pointwise fixed.) It is a consequence of a, and b, that the set of fixed points under t form a subcomplex L of K. We will denote by k the quotient space K/t, formed by identifying points in K which are t images of each other. p shall denote the projection K -k and we set'l = p(L).

to its limit F (y) was considered with respect to its uniformity in y. Ill that paper certain conditions are given which are sufficient to insure the uniform convergence of the sum (A) with respect to y, and that independent of the choice of Ui on each interval. An example at the end of the paper shows that the convergence need not be uniform in y independent of the choice of Us even though f(x, y) be bounded and continuous in each variable separately. This example points the way to a further sufficient condition, and in the case of a certain class of functions, to a condition which is both necessary and sufficient. The first part of the paper consists in the development of these conditions. In the second part of the paper we proceed to a discussion of the continuity of a function defined by a Riemann-Stieltjes integral, and the uniform convergence of a RiemannStieltjes sum similar to (A). Part I.

This paper is a continuation of two other papers with the same title [1, 2]. Let GLn[R] be the group of n x n invertible matrices with elements from a ring R. In [1], R was the ring of integers modulo a prime power pr = v, and the normal subgroups of GLn[R] were described. For each ideal W in the ring R, two normal subgroups N*,n, Nvn were defined by means of congruences: Nvn {M det M = 1, M-I, identity, mod W}; N*,n = { M M mod W is scalar}. It was shown that every normal subgroup of GLn[R] (n>2) contains Nw n and is contained in N*,n for suitable W. This result was generalized in [8] to the case where R is any valuation ring. The important case where R is the ring of rational integers, and GLn[R] is the unimodular group, is not covered by either of these references. A study of the latter case is the primary purpose of this paper. The methods used here work reasonably well when R is any euclidean ring (even a non-commutative one); certain results hold in a still wider class of rings. The general problem of describing normal subgroups falls naturally into two parts. In the first part (?? 2, 3) the discussion is restricted to the case n > 2. There we show that if a normal subgroup N contains a certain matrix C, then N must contain also the canonical matrix A =I+ me2l for some positive integer m. The second part (?? 5, 6) includes the case n = 2. The problem here is to find the least normal subgroup Qmn of GLn[R], or of the subgroup SLn[R] of the latter, which contains A. For R = J, we conjecture that Qm,n is Nm,n = N(m),n, and establish the validity of this conjecture for m 2) into a form parallel to and generalizing the solution given in [1]; v need not be a prime power (? 4). The results of [8]

1. In a former paper* I have discussed the conditions to be imposed on a finite continuous group with a single exceptional infinitesimal transformation in order that some of the structural constants may be simplified. It is the object of this paper to extend these discussions to groups with any finite number of exceptional infinitesimal transformations. 2. Let the group G be generated by r + g (r, g integers) infinitesimal transformations, and let their corresponding operators be X1, X2, *,X, Xr+ll 1** Xr+g where

We discuss in this paper the algebraic and transcendental extensions of completely primary rings. See the introduction of the author's paper "Completely Primary Rings I", published in this journal and referred to as CPI, for an introduction to the present paper. We continue the notations and conventions of CPI. Hence, the term ring always denotes a commutative, nonzero ring with unit element 1. If S is a ring, N(S) denotes its radical and H the natural homomorphism from S onto the residue class ring S/N(S) = S. If B is a subset of S B C S denotes the image of B under H. If B is a subring R of S, R is identified with R/N(R) and the contraction of H on R is considered as the natural homomorphism from R onto R/N(R).

The purpose of this paper is to develop the theory of elementary divisors, to prove the approximation theorem, and to determine the class number for the following two types of algebraic groups: (i) the unitary group of a hermitian form over an algebraic number field; (ii) the unitary group of a hermitian form over a quaternion algebra, having an algebraic number field as center, with respect to the canonical involution. The latter includes the symplectic group as a special case, since we admit the total matric algebra of degree two as a quaternion algebra. Of course, the approximation theorem is obtained only in the case where the hermitian form is indefinite. A problem of the same kind was investigated for the group of regular elements of a simple algebra by Eichler [5], and for the orthogonal group by Eichler [6, 7] and Kneser [9]. In [11], we treated the groups of type (ii) with the same intent and obtained a somewhat weaker result than in the present paper. We now explain our result by giving a summary of each section. Let F be an algebraic number field of finite degree and K a quadratic extension of F. Let V be a vector space over K of dimension n and ?F(x, y) a non-degenerate hermitian form on V with respect to the non-trivial automorphism of K over F. We denote by U( V, F) and G( V, F) respectively the unitary group of iF and the group of similitudes of iF (cf. 1.1 below). Further, we denote by SU(V, (D) the subgroup of U(V, iF) consisting of the elements with determinant 1. If n is even, we can consider the group of direct similitudes H( V, iF) (cf. 2.2). In the first two sections, we give preliminaries for these four groups, and study elementary properties of maximal lattices, whose definition is as follows. Let g and x be the ring of integers in F and in K, respectively. Let L be an x-lattice in V. We denote by Me(L) the ideal in F generated by the elements iF(x, x) for all x e L, and call it the norm of L. We say that L is maximal if L is a maximal one among the x-lattices with the same norm. Let p be a prime ideal of F, and F, the completion of F with respect to P; and let K, = K ?F Fp and V, = V OF Fi. In ?? 3 and 4, we treat the local theory of elementary divisors for the lattices in V,. If p decomposes in F, KP is isomorphic to F, x F,, and therefore things are much easier. Section 3 is concerned with this case. In ? 4, we discuss the case where K, is a field. 369

I am indebted to George Kolettis, Jr., for pointing out that the "if" part of Proposition 5.2 is incorrect. A regular group is not necessarily a direct sum of rank one groups. The proof of the converse is unaffected. A counter example can be found in his paper "A theorem on pure submodules" which will appear in the Canadian Journal of Mathematics. The mistake appears on line twelve from the bottom on page 388, where it is incorrectly reasoned that n * h(m, p) + h is p-power divisible. None of the other results of the paper are affected.

In a recent series of short papers [2, 3, 41 I have been discussing the relationships of uniform convexity with certain other properties of normed vector spaces. In this paper I propose to discuss relationships between uniform convexity, factor spaces, and conjugate spaces. Because of the large number of special results which are needed for two dimensional spaces-and which are false in general-the paper will be divided into two parts: In Part I (??2-4) B is two dimensional; in Part II (??5-7) this restriction is removed and B may be any normed vector space. To recall the definition, B is said to be uniformly convex if there exists a function 6 such that 0 e, is a modulus of convexity for B. Note that if 51 is nowhere greater than 5, a modulus of convexity for B, then 5l is a modulus of convexity for B. Because of the pointwise nature of the definition of uniform convexity, it is clear that B is uniformly convex with modulus of convexity a if and only if all the two dimensional subspaces of B have the common modulus of convexity 6. One of the most useful results of this investigation (Theorem 5.5) is the complementary fact that B is uniformly convex if and only if all the two dimensional factor spaces of B have a common modulus of convexity. In the study of the effect of uniform convexity of B on the nature of the conjugate space B*, this result makes it possible to reduce the problem to the study of two dimensional spaces. In such a space B uniform convexity is equivalent to strict convexity: that is, a two (or finite) dimensional space is uniformly convex if and only if there does not exist a line segment of positive length all of whose points are of norm one. It has been observed [1, Footnote 13] that such a line segment on the unit sphere is equivalent to the existence of a "sharp edge" on the unit sphere in B*. The attempt to describe a "sharp edge" of the unit sphere in a finite dimensional space B in terms of the norm in that space leads to the condition that there exists a k > 0 such that for any e > 0 a pair of points bi and b2 exists such that II b -b2 11 k 11 bl-be 11 . Contradicting this suggests the following condition, a sort of dual concept to uniform convexity. A space B is said to be uniformly flattened if there exists a function v7 positive for 0 < e ? 2 such that lim.b.o 7(e) = 0 while (211 b + b2 11)/i b2 11 < 7(e) if J]b11 = JJ b1 = 1 and 11 bb2 ? || ; 375

Urysohn [4, p. 290] has considered the problem of determining the most general class of topological spaces in which non-constant real-valued continuous functions exist. He showed that a countable Hausdorff space exists in which every continuous real-valued function is constant [4, pp. 274-283]. Pospisil [2, pp. 1-3] has extended this result by exhibiting a Hausdorff space of arbitrary infinite cardinal number in which every continuous real-valued function is constant. The spaces constructed by both Urysohn and Posp ?il, however, fail to satisfy the Urysohn separation axiom.2 Urysohn posed, but left unanswered, the question of whether or not regular topological spaces exist in which every continuous real-valued function is constant. In the present paper, it will be proved that regular spaces exist in great profusion on which every real-valued continuous function is constant. Urysohn also considered the problem of determining the least cardinal number of a connected space. [4, pp. 274-283], and, by the construction mentioned above, showed that a countable connected Hausdorff space exists. This example, however, does not satisfy the Urysohn separation axiom. Urysohn's result is extended in the present paper by the construction of a countable -Urysohn space in which every continuous function is constant and which is accordingly connected. THEORBM 1. Let N be an infinite cardinal number which is not the sum of No cardinal numbers smaller than R. Then there exists a regular space 'k of cardinal number N in which every continuous real-valued function is constant. Our construction of the space ?k, which proceeds in several steps, makes essential use of a space constructed by Tychonoff [3, pp. 553-556] and the condensation process devised by Urysohn [4, pp. 274-283]. First, let A be the least ordinal number with corresponding cardinal number N. It is obvious that A is not the limit of a countable sequence of ordinal numbers smaller than A. Let D be the set of all ordinal numbers a such that 1 ? a < A Let C be the set of all ordinal numbers a such that 1 < a ? c, where X denotes, as usual, the least infinite ordinal number. D and C are made into topological spaces by the following construction. In D, every point 6, where 1 < a < A, is isolated. Neighborhoods Uy(A) of the point A e D are the sets E[b; y < _5 ?A]. C is given a similar neighborhood system with X as the only non-isolated point. Let S be the set of all pairs (6,a) of ordinal numbers such that ( E D and a E C.

As far as the ends of a space or of a group are concerned see H. Freudenthal [3]1, [4], H. Hopf [5] especially No. 16, p. 96 and E. Specker [12] p. 320. See also ?3 of this paper. All 3-dimensional manifolds in the present paper are connected and are supposed to be considered with a fixed triangulation. This is not at all a restriction of generality according to E. E. Moise's work [8]. In the present paper S means the 3-sphere and K a polygonal knot in it. In his thesis E. Specker [12] p. 329 proved the following: The conjecture that the knot space S K is aspherical is equivalent to the conjecture that the knot group 7ri(S K) has one or two ends. So naturally the question arises: When has iri(S K) one and when does it have two ends? It seems to be reasonable that the answer should be: 7r,(S K) has two ends if and only if K is algebraically unknotted, i.e., 7r1(S K) is free cyclic. The present paper developed from an attempt to prove asphericity for knots and to clarify when a knot group has one or two ends. Unfortunately a solution of the problem of asphericity of knots escaped us. But on the other hand the basic character for our work of this problem is apparent, for in order to clarify when a knot group has one or two ends it is necessary to assume asphericity. The main result of this paper is: If asphericity of knots holds, then a knot group has one or two ends according as the knot is algebraically knotted or unknotted. This result is an easy consequence of Theorems 1 and 2, and is proved in ?4, where also the different notions of knottedness are explained. As for Theorem 2, which is explained in ?3, we have to notice that its proof is based on two theorems of J. H. C. Whitehead contained in his papers [15] and [16]. However, in proving Theorem 1, which is also explained in ?3, a preparatory work was needed. The proof is based on Corollary 2, explained in ?2, and the main lemma, explained in ?3. Both are based on Lemmas 1 and 2, and Corollary 1, explained in ?2. The author of the present article would like to express his gratitude to the Mathematical Department of Princeton University, which, by giving him the privileges of a Visiting Fellow, enabled him to use the facilities of the Fine Hall Library and to attend the lectures of the Department, especially those of Professors R. H. Fox and N. E. Steenrod. The author is especially indebted to Professor R. H. Fox whose experience on 3-dimensional space was most valuable. In the ??2 and 3 of the present paper some simplifications, suggested by Professor R. H. Fox, have been incorporated.

In order to describe the contents of the present paper we begin with some definitions. An extension of a local group Y is a pair (E, 4) where E is a local group and-? is a strong homomorphism of E onto X. Let (E, 4) be an extension of Y. It follows from the definition of strong homomorphism (?31) that the set N = 0_1 is a subgroup of E; N will naturally be called the kernel of (E, 4). A function u on Y to E such that qu(y) = y is a selector of (E, 4). A selector u of (E, 0) is symmetric if u(y'1) = u(y)V'. A selector u(y) is multiplicative if u(yiy2) = u(yI)u(y2) whenever YiY2 is defined. (Since the homomorphism 4 is strong, u(y1)u(y2) is defined whenever YiY2 is defined.) The extension (E, 0) of Y is inessential if it admits a multiplicative selector. Let Y be a local subgroup of a group Q generated by Y. We call an extension (E, 0) of Y extendible over Q if E can be imbedded in a group which is mapped onto Q by a homomorphism whose contraction to E is 0. Assume now (and during the remainder of the introduction) that Q is simply connected relative to Y (?34) and that Y contains no elements of order 2. Consider the abelian group P2(Y) defined in ??13, 34. Although it is determined by Y alone, P2(Y) is isomorphic to the second homotopy group of the complex K(Q, Y) (?35). The main theorem in the present paper is the following: in order that every extension of Y be extendible over Q it is necessary and sufficient that P2(Y) = { 1}. We leave open the question of the relationship between extendibility and the structure of P2(Y) when the latter is non-trivial. Consider the natural homomorphism aQY:G(Y) -4 Q defined in ?34. (G(Y) is a free group since Y has no elements of order 2.) Let C = C(Q, Y) be the preimage of X under aQY ; C is a local subgroup of G(Y). The problem of determining whether an extension of Y is extendible over Q is closely related to the problem of determining whether a given extension of C is inessential. We shall show that every extension of C is inessential if p2(Y) = {1}. Since the proof gives a somewhat misleading appearance of complexity, it will be well to outline the steps. It will be shown that the local subgroup C of G(Y) contains the canonical generators of G(Y) (?32) as well as the group R of defining relations of Q (?34).

Eilenberg in 1944 ([E])2 removed some of the difficulties associated with the classical concept of "singular cell" and obtained a new singular homology theory. In 1948, Hurewicz, Dugundji, and Dowker ([H]) applied the methods of inverse and direct limit groups in conjunction with mappings of polytopes into a space to obtain another singular homology theory. The primary purpose of this paper is to show that these two methods yield isomorphic cohomology (homology) groups. We prove this result by constructing a simplicial polytope P(X) which is a geometric realization of the Eilenberg singular complex of the space X, and then showing that the n-cohomology group of P(X) is isomorphic to the n-cohomology groups of X as defined by Eilenberg and by Hurewicz, Dugundji, and Dowker. We also prove that the Hurewicz homotopy groups of P(X) and the space X are isomorphic. The above results are stated in full in Section 6. Section 6 contains the statements of the principal results of this paper, although Theorem I is given in Section 5 and Theorem VII is given in Section 10. In Sections 7 to 9 we prove the results of Section 6. In Sections 2 to 4 we briefly summarize the definitions and theorems from the papers of J. H. C. Whitehead ([W]); Hurewicz, Dugundji, and Dowker ([H]); and Eilenberg ([E]) which are needed in the remainder of this paper. The basic notation used will be that of [HW]. Beyond that, our notation will be consistent with [H], and, in regard to the Eilenberg complex, [E]. We will use to denote group isomorphisms, and to denote homotopy relations. The topological spaces and groups considered here are of the most general type with no separation axioms postulated. We will use infinite cochains over a topological, abelian coefficient group. Chains will be finite and with a discrete, abelian coefficient group. The word simplex will always mean a closed geometric (Euclidean) simplex unless we explicitly state otherwise. We will let iz-n(Y, y) denote the Hurewicz n-homotopy group of the space Y with the base point

A link projection' is said to be alternating iff it is connected and, as one follows along any component of the link, undercrossings and overcrossings alternate. A projection is trivial iff it is connected and has no crossings; otherwise it is non-trivial. We include trivial projections as alternating. An alternating link type is one which has an alternating projection. The principal result of this paper is Theorem 3.5 which contains the assertion that the degree of the reduced Alexander polynomial of an alternating link type plus one equals twice its genus plus its multiplicity. A second result, obtained as an immediate corollary of the same method which ultimately yields (3.5), is Theorem (2.13): The reduced Alexander polynomial of an alternating link type is an alternating polynomial. This theorem provides the simplest proof of the existence of non-alternating types. The image P of a connected, non-trivial projection has a natural decomposition as a graph. The vertices are the crossings, i.e., the images of the undercrossings, and the edges are the open arcs into which the crossings subdivide P. Since we shall have no reason to distinguish a point at infinity, we regard P as a spherical rather than a planar graph. The results of this paper are obtained by studying the image graph of a non

Suppose X and Y are topological spaces, and H(X) is the group of all homeomorphisms of X onto itself. We are concerned, in this paper, with the following question: (a) if p is a group isomorphism between H(X) and H(Y), does there exist a homeomorphism w of X onto Y such that (p(h) = h-1 for all h e H(X)? A more restricted question, which is also of interest, would be: (b) is every group automorphism of H(X) inner? The answer to (a) is, in general, no, for we may let X be the closed interval [0, 1], Y= (0, 1), and, for each h e H(X), 9(h) be the restriction of h to Y. To show that (b) also has a negative answer in general, let X = [O, 1] U (2, 3), w be a homeomorphic mapping of (0, 1) onto (2, 3) and (2, 3) onto (0, 1), and 9(h) be the extension of wha-v to X. Then 9 is an outer automorphism of H(X), for w can not be extended to an element of H(X). If X and Y are discrete spaces, then the answer to both questions is known to be yes, except in (b) when X has six elements (cf. [2] for a short historical survey). Question (a) has been studied in a more general setting by Gerstenhaber [2]. Fine and Schweigert [1] proved that (b) has an affirmative answer when X is a 1-cell, either open or closed, and Gerstenhaber [4] proved the same when X is a closed 2-cell. In the present paper, we will show that (a) has an affirmative answer when X and Y are compact, locally euclidean manifolds, with or without boundary. By setting X= Y, we also obtain an affirmative answer to (b) for the same class of spaces. The results obtained actually show that an isomorphism between any two sufficiently large subgroups of H(X) and H( Y) implies that X and Y are homeomorphic. In the course of this work, we develop a representation for the minimal normal subgroups of any sufficiently large subgroup of H(X).

Two simplicial complexes, Ko and K1, are called isomorphic if their respective sets of vertices can be so numbered, Pi and Qi (i = 1, 2, ), that Qi, ... Qim is a cell of K1 when and only when Pi. * * * Pi,,, is a cell of Ko. We will then say that the vertices are similarly numbered. A complex on a euclidean 2-sphere will be referred to as geodesic if each of its 1-cells is an arc of great circle shorter than a semi-circle and each of its 2-cells is a spherical triangle less in area than a hemisphere. Every complex throughout this paper will be finite and will be non-singular, in the sense that no two different cells of any one complex will have a point in common. DEFORMATION THEOREM. Let Ko and K1 be a pair of isomorphic geodesic complexes on a euclidean 2-sphere, S. Let Pi and Qi (i = 1, * , n) be the verticeo, similarly numbered, of Ko and Kt respectively. If and only if the isomorphism3 between Ko and K1 can be extended into an orientation-preserving self-homeomorphism of S, it is possible to define, for every t (0 < t < 1), a geodesic complex, K1, with vertices Pi(t) (i = 1, * * *, n) in such a way that (1) Pi(0) = Pi and Pi(1) = Qi (i = 1, * * *, n) (2) K1 and Ko are isomorphic with vertices similarly numbered (3) as t increases from 0 to 1, Pi(t) traces a continuous curve on S (i = 1, * , n). The conclusion of this theorem asserts the existence of an isotopic deformation of Ko into K1 induced by a motion of the vertices, all cells remaining geodesic and non-degenerate throughout the motion. The proof of the theorem will occupy most of the remaining sections of the paper. Before commencing the proof, we state a criterion for the fulfilment of the necessary and sufficient condition involved in the theorem.

Let X be a topological space with A c X a subspace, and let z-: (X, A) (X, A) be an involution; i.e., a continuous map z-: X X with square the identity and such that MA c A. Combining the notions of bordism (Atiyah [1]) and of differentiable periodic maps (Conner and Floyd [4]), one may define bordism groups of the involution (X, A, z). Specifically, a (free) equivariant bordism class of (X, A, z) is an equivalence class of triples (M, A, f) with M a compact differentiable manifold with boundary, A: M e M a differentiable (fixed-point free) involution on M, and f: (M, AM) (X, A) a continuous equivariant map [zf = fte] sending AM into A. Two triples (M, a, f ) and (M', a', f') are equivalent, or bordant, if there is a 4-tuple (W, V, v, g) such that W and V are compact differentiable manifolds with boundary, a V = AM U AM'and a W = M U M' U V/DM U AM' a V, v: (W, V) (W, V) is a differentiable (fixed-point free) involution restricting to , on M and ,t' on M', and g: (W, V) (X, A) is a continuous equivariant map [zg = gv] restricting to f on M and f ' on M'. The disjoint union of triples induces an operation on the set of (free) equivariant bordism classes of (X, A, z) making this set into an abelian group. This is a graded group, where the grading is given by the dimension of the manifold M, and one lets 9Z,(X, A, z) be the group of equivariant bordism classes of (X, A, z), and one lets 9Z*(X, A, z) be the group of free equivariant bordism classes of (X, A, 4). If A is empty, one writes W*(X, z) and 9M(X, z) for these groups. Letting X be a point, with A empty and z the identity map, this reduces to the situation studied by Conner and Floyd [4], with 9Z*(pt, 1) = I*(Z2) and 9Z*(pt, 1) = 9*(Z2) in their notation. The main purpose of this paper is to compute the groups 9Z*(X, z) and 9Z*(X, z) and to explore their interrelationships. As always, however, the study of an involution (X, z) is really the study of a pair (X, FT, z), with FT the fixed-point set of z, and one is forced to consider pairs in order to study the absolute case. To study more general pairs tends to force a study of 4tuples, and this additional complication will naturally be avoided. The major portion of this paper is a geometric analysis of bordism of in

The principal purpose of this paper is to construct in Euclidean space R n, n > 3, a compact zero-dimensional set, A, whose complement is not simply connected. The construction constitutes an affirmative answer to a problem proposed by Borsuk [1], and is a generalization of Antoine's set [2] for the case n = 3. The paper is divided into three parts. In Part 1 the set A is constructed and proved to be zero-dimensional. In Part 2, 7rl(S nA) is computed explicitly and represented non-trivially2 in the symmetric group S6 . In Section 3 it is shown that the construction cannot be accomplished in Hilbert space. More precisely, it is shown that the complement of any compact subset of Hilbert space is contractible. I have been unable to determine whether a set of finite dimension can leave the Hilbert cube multiply connected, but have shown, however, that if there be such a set, then for some n its projection in the subspace xi = x2 = * = Xn = 0 is of infinite dimension. An analogous result is shown to hold for subsets of Rn; namely, the projection of such a set in any (n 1)-dimensional hyperplane has positive dimension. An interesting corollary to this, pointed out to me by E. E. Floyd, is the fact that the set A C Rn C Rn+l leaves Rn+l simply connected. Finally in Section 3, a remark by J. WV. Alexander [4] on Antoine's construction is generalized to show that for each 1 < q < n there is a q-cell in Rn whose complement is not simply connected.

Given two sphere bundles (5 and 52 over the base complexes K1 and 1K2 respectively, it is possible to define in a natural way a "product bundle" over the product complex K1 X K2 . When K1 = K2= K (say), the part of the product bundle over the diagonal of the product complex K X K is the product bundle in the sense of Whitney.' We shall prove in the present paper that a certain duality theorem holds for the product bundle over K1 X K2 and that Whitney's duality theorem for sphere bundles follows from this more general duality theorem as a consequence. (Throughout the paper coefficients mod 2 will be used.) The idea of this proof seems to be quite different from Whitney's original one, of which only a brief sketch is known.2 The paper is divided into three sections. In ?1 some preliminary considerations and theorems on vector fields are given. A duality theorem for the product bundle over K, X K2 is then proved in ?2. ?3 is devoted to a proof of Whitney's duality theorem.

1. Let SL(n, Z) be the group of all n x n-matrices with rationali nteger coefficients and det = + 1. In the present paper, we shall study the finite factor groups of SL(n, Z). For n = 2, every finite group which can be generated by two elements of order two and three is a factor group of SL(2, Z). It has long been conjectured that the situation is quite different for n > 2. In the following, we shall assume n > 2 unless otherwise stated. In a recent paper [1] in this journal, J. L. Brenner has established the following ladder relation. Let N be any normal subgroup in SL(n, Z), not being trivial, i.e., not being the identity subgroup or the center. Take any matrix of N, and let ja be the g.c.d. of the non-diagonal coefficients and the differences of diagonal coefficients. Let m be the g.c.d. of the numbers Pa for all elements of N. Denote by Qn,,7 the least normal subgroup of SL(n, Z) which contains I + me21 (the notation being as in [1]), and let Nm*,. be the normal subgroup of all matrices which are congruent to a scalar matrix modulo m. Then the following ladder relation holds

The theory of analytic varieties of several complex dimensions (more than one) as distinguished from the theory of Riemann surfaces, contains no general theorems of classification and existence. It is known that the direct generalizations of Riemann's theorems concerning the existence of non-trivial meromorphic functions and Abelian differentials are false for the several variable case. Further the identification of compact Riemann surfaces with algebraic curves has no analogue in several variables. Riemann's original account of his theorems was founded upon a variational method, the celebrated Dirichlet Principle. In this paper we shall generalize this variational method, that is, we formulate a variational problem whose extremals are analytic functions of several complex variables. We give a necessary and sufficient condition for the existence of a non-trivial bounded complex analytic density on an arbitrary complex analytic manifold. The restriction to boundedness (which is defined later on) is superfluous for compact manifolds. Our argument makes no distinction between the one variable and several variable case. The formulation of our result is given in terms of Hilbert spaces as suggested by Weyl's paper on the method of orthogonal projections. We associate with an arbitrary complex analytic manifold V, two Hilbert spaces V3* and 3*. 3* is a closed subspace of .3** The space 8* is given by an explicit construction, while 23* is defined indirectly. The main theorem asserts that the orthogonal complement of S* in V3* is precisely the space of bounded comlpex analytic densities on V. There is a vacuous element to this result; that is, nothing is asserted concerning the proper inclusion of 3* in p3*. This was to have been expected as an arbitrary complex manifold does not necessarily have any non-trivial complex analytic densities. For the case of Riemann surfaces one settles the question of the existence of non-trivial Abelian integrals of the first kind with the aid of topological considerations. There are no known topological conditions, applicable to arbitrary higher dimensional manifolds, that would assure the existence of non-trivial complex analytic densities. 1. V is a complex analytic manifold of n complex dimensions. DEFINITION 1. Suppose we have assigned to each pair (p, (ui, . , u,)) consisting of a point p of V and an ordered set u1, .. *, u of complex coordinates of V centered at the point p, a complex valued function fp,(u) defined in some nbd. of p, in such a fashion that

Referring to the next section for any unfamiliar notation or definition, let us consider the following statement. If m > 0 n > 1, R is an open subset of En , and f is a function on R to El of class Cm then f transforms its critical set into a set of (linear) measure zero. H. Whitney2 has shown the statement false if m = n 1 > 1; M. Morse and A. Sard, in an unpublished paper, have shown, on the other hand, that it is true providing m is the greatest integer in n + (n 3)2/16. This last result contains as a special case the fact that the statement is true in case m = n = 1, 2, 3, 4, 5, 6. Thus it is natural to ask: Is the statement true in case m =n? Theorem 4.3 of the present paper answers this question in the affirmative. As a matter of fact the answer is yes under the less restrictive assumption that f be of class C(m1-) providing each partial derivative of order (m 1) is totally differentiable. We give here no proof of this last statement; of the two proofs with which we are familiar one involves a fairly comprehensive modification of Section 3 as well as an additional lemma while the other involves some properties of an (n 1) dimensional measure of sets contained in E. . An obvious corollary of Theorem 4.3 is Theorem 4.4 which states that a function on an open subset of En to El which is of class C' is constant on any connected subset of its critical set. Theorems 4.3 and 4.4 are fairly direct consequences of Theorem 4.1 which is itself closely related to Theorem 3.7. Theorems 3.7 and 4.1 really form the kernel of the paper. We believe Theorem 4.1 in particular is susceptible of wider application than herein given.

uniquely,-although F(xi, X2, ... ) may in general be multiple-valued. This condition (2)-for a finite set of numbers-was used by 0. Sutol as one of three conditions to characterize the arithmetic mean; and it has been used by several other writers. It will not be required that the mean m be intermediate between the least and the greatest of the x's-or of their lower and upper bounds. Indeed, 0. Chisini2 and B. de Finetti3 have given illustrations of "external" means. And the mean in Illustration 3, which I give in a later section of this paper, is of this sort. Intermediacy or internality ceases to be of great importance if emphasis is laid upon the use to which the mean is to be put. If the condition of intermediacy is satisfied, then (2) is satisfied; but the converse of this statement is not true. For the theorems of this paper, the uniqueness of F = x in (2) will be adhered to strictly; but a useful extension of the definition of a mean-which merely requires that at least one value of F(x, x, * , x) shall be x-will be exemplified in the first part of Illustration 5. Definitions of certain properties of means-following De Finetti-will now be given for means of a finite or countable set of elements. The accompanying abbreviations will be used subsequently either for the adjective or for the corresponding noun. The symbol Max will stand for the maximum element or the least upper bound of the elements, and Min for the least element or the greatest lower bound of the elements.

A transformation of a metric space which changes distances by less than a certain fixed positive number e is called an approximate isometry, or an eisometry. In a preceding paper2 it was shown that an e-isometry taking Hilbert space into itself could be uniformly approximated within a distance of ke by a true isometry, where k was a positive constant not greater than 10. The object of the present paper is to establish a similar "stability" theorem for transformations of the space of continuous functions over a compact metric space. According to a theorem of Banach,8 every isometry between the normed spaces E and E' of continuous, real valued functions over the compact metric spaces K and K', respectively, is generated by a homeomorphism between K and K', and conversely. This result suggests a method of attack on our stability problem. The first step is to prove the existence of an isometry between the spaces E and E', having given the approximate isometry between these spaces, and this can be done by showing that the underlying spaces K and K' are homeomorphic. However, the method used by Banach in establishing the correspondence between the points of K and K', which was based on a theory of peak functions, seemed rather difficult to apply to our problem. Therefore a different method, dealing with the hyperplane of all functions having the same value at a given point of K, is employed in the present paper. The norms in the spaces E and E' will be defined in the usual way, as the maximum of the absolute value of the function in question. A transformation T(f) taking E into E' will be called an e-isometry if II T(f) T(g) II If gII I 0 there exists a point q of K' and a redl number c such that T(M(p b, a)) C M'(q, c, a + 3e/2). PROOF. Without loss of generality, we may assume that the metrics in K and K' have been chosen so that the diameter of each of these spaces is one. We shall assume for the present that a > 0, and consider the case a = 0 later. Putfn(x) = b + 3n np(x,p), gn(X) = b 3n + np(x, p), for n = 1, 2, 3, * * and x in K, where p(x, p) denotes the distance from x to p. Consider the spheres Sn = [kp; II (p-fIn I _ 3n + a], R. = [(p; I I -gn I I _ 3n + a] in the space E

where xa are integral, m is a given integer > 1, c is a given real but not an integral number > 1, a tends to zero and a tends to unity when N increases infinitely. In the papers mentioned we proved that it is possible to find values of r depending on c or on m, but not on N, such that every real number N. > than a certain constant No, can be represented in the form (1) or (2). Let G(c) be the least r such that every N > No can be represented in the form (1) and H(m) be the least r such that every N > No can be represented in the form (2). We obtained in the papers mentioned the following relations for the numbers G(c) and H(m)